Question

2. Using substitution to simplify a problem (a) Solve the following (homogeneous) differential equation using the appropriate substitution. (b) Find the solution to the equation T+3 Hint: The same substitution wil no longer work, but the equation is almost homogeneous. Use a substitution of the form r- X - h, y-Y - k to reduce this problem to the problem solved in part (a), i.e. choose h and k so that this problem becomes homogeneous in the substituted variables X and Y

Answer 1

put y = zx in given equation to reduce it into x and z solve for z and replace z by y/x for part (b), put x = x - h, y = y - k in equation choose h and k such that resulting equation is homogeneous use the solution of part (a) to get its solution given difference equation is dy/dx = x + 2y/x -(1) therefore R in R H S, numerator and denominator are homogeneous function of same degrees therefore given differential equation is homogeneous therefore we put y = zx dy/dx = dz/dx.x + z.1 dy/dx = x dz/dx + z putting above values in (1), we get, x dz/dx + z = x + 2zx/x x dz/dx + z = x(1 + 2z)/x x dz/dx + z = 1 + 2z x dz/dx + = z + 1 dz/z + 1 = dx/x ln(z + 1) = lnx + lnc ln(z + 1) = ln(xc) z + 1 = xc on integration